Given a quantized enveloping algebra U returns the map
from U onto the integral form of the universal enveloping algebra
of the corresponding Lie algebra (cf. Section The Z-form of Uq(L)).
We refer to Section Universal Enveloping Algebras for an account of universal enveloping
algebras in Magma.
> U:= QuantizedUEA(RootDatum("C3"));
> f:= QUAToIntegralUEAMap(U);
> p:= CanonicalElements(U, [1,2,1]);
> [ f(u) : u in p ];
[
y_1*y_2^(2)*y_3,
2*y_1*y_2^(2)*y_3 + y_1*y_2*y_5,
y_1*y_2^(2)*y_3 + y_1*y_2*y_5 + y_1*y_7,
y_1*y_2^(2)*y_3 + y_2*y_3*y_4 - y_2*y_6,
2*y_1*y_2^(2)*y_3 + y_1*y_2*y_5 + y_2*y_3*y_4 - y_2*y_6 + y_4*y_5,
2*y_1*y_2^(2)*y_3 + y_1*y_2*y_5 + 2*y_2*y_3*y_4 - y_2*y_6 + y_4*y_5,
y_1*y_2^(2)*y_3 + y_1*y_2*y_5 + y_2*y_3*y_4 + y_1*y_7 + y_4*y_5 + y_8
]
So this allows one to construct elements of the canonical basis of
a universal enveloping algebra (of a semisimple Lie algebra).
V2.28, 13 July 2023